Resolving a vector into Components

The following figure shows a vector \(\overrightarrow {AB} \) of length \(5\sqrt 2 \) units, inclined at an angle of 450 to the positive x-direction:

Can we express this vector using the \(\hat i\)- \(\hat j\) system?

Observe the following figure, where we have resolved the vector \(\overrightarrow {AB} \) into two components \(\overrightarrow {AC} \) and \(\overrightarrow {CB} \), where \(\overrightarrow {AC} \) is parallel to the x-axis, and \(\overrightarrow {CB} \) is parallel to the y-axis:

Let us see another example of resolving a vector into its components (which simply means finding the \(\hat i\)-component and the \(\hat j\)-component). The following figure shows a vector \(\overrightarrow {PQ} \) of length 6 units inclined at an angle of 1200 to the positive x-direction:

What will be the components of \(\overrightarrow {PQ} \)?

Observe the following figure:

\(\overrightarrow {PR} \) is the \(\hat i\)-component of \(\overrightarrow {PQ} \), while \(\overrightarrow {RQ} \) is the \(\hat j\)-component of \(\overrightarrow {PQ} \). We have:

Example 2: A force of magnitude F is acting on a box at an angle of \(\theta \)to the horizontal. Take the horizontal right direction as the \(\widehat i\) direction, and the vertical up direction as the \(\widehat j\) direction. Specify the force \(\overrightarrow F \) using this \(\widehat i\)- \(\widehat j\) system.

Solution: The vector components of the force \(\overrightarrow F \) are shown below: